Circles and Angles - Home Page -- Tom Davis

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Circles and Angles
Tom Davis
tomrdavis@earthlink.net
http://www.geometer.org/mathcircles
January 10, 2000
Here are the key theorems relating circles and angles.
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Figure 1: Central Angle and Inscribed Angle
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The diagram on the left of Figure 1 gives the definiton of the measure of an arc of a
circle. The arc from to is the same as the angle . In other words, if ,
then the arc also has measure .
On the right of Figure 1 we see that an inscribed angle is half of the central angle. In
other words, .
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Figure 2: Right Angle, Difference of Angles
On the left of Figure 2 is illustrated the fact that an angle inscribed in a semicircle is
a right angle. On the right of Figure 2 we see that an angle that cuts two different
arcs of a circle has measure equal to half the difference of the arcs. In other words,
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Figure 3: Inscribed Quadrilateral, Tangent
Finally, in Figure 3 we see that if a quadrilateral is inscribed in a circle, then the opposite
angles add to , and conversely, if the two opposite angles of a quadrilateral
add to , then the quadrilateral can be inscribed in a circle.
On the right, we see that a tangent line cuts off half of the inscribed angle:
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